Karen adds: This was very much a group effort and these contributors (listed in reverse alphabetical order because regular alphabetical order feels a bit tyrannical when imposed by an Abbott) each made this post significantly more interesting than anything I would have come up with on my own – thanks to all.

*******************

Over the past five years or so, I have spent more time than I should probably admit feeling hung up about what stochasticity really means. When my thoughts start to fall down a rabbit hole of semantics or to philosophical questions of determinism versus free will, I pull myself back. I’m not interested in those things, important as they may be. I’m interested in gaining a deep understanding of the role that “stochasticity” — the conceptual construct — plays in ecological thinking, as well as the role that actual stochasticity plays in real ecological systems.

Lots of other people think about these things too (particularly the latter question on the role of actual stochasticity), so I asked a non-random group of colleagues, collaborators, and lab members to share their thoughts on what stochasticity means and where or how stochasticity is important. I’m not sure if this exercise has made me feel more or less hung up, but it was really fun and a number of interesting themes emerged:

1) Collectively, we have lots of ideas about what stochasticity is, and what it’s not. These ideas can be roughly organized by whether stochasticity contributes as a mechanistic driver of observable patterns or whether it exists outside of (deterministic) drivers to add variance to observations.

2) Questions of scale are ubiquitous.

3) Semantics aside, there are some meaningful differences in how ecologists apply the concept of stochasticity. “Stochasticity” is a rather precise-sounding, technical word that may hide a lack of conceptual precision.

Taking these one at a time —

1) What stochasticity is, and what it’s not:

There was universal consensus that stochasticity involves uncertainty, but a bit of divergence on the origin of that uncertainty. I tend to use the word “stochasticity” synonymously with “randomness” and “noise” and many contributors also defined stochasticity as randomness. Detractors on this point questioned whether anything in ecology is truly random. If we had perfect knowledge of all causal factors, would there ever be more than one possible outcome? An equal but opposite argument is: since we never can have perfect knowledge, is anything in ecology truly deterministic? There’s that pesky rabbit hole again. So, sidestepping that, we could say that stochasticity is anything with a probabilistic aspect, defined by not only a mean but also a variance and distributional shape. The uncertainty associated with stochasticity then comes from randomly sampling from this distribution. (Additional uncertainty might come from not fully understanding the distribution itself, but I wouldn’t consider that to be stochasticity, per se, because it doesn’t stem from randomness. Some of my colleagues might disagree here, though, as per the next paragraph.)

What if nothing in ecology really is random? In this worldview, stochasticity is a stand-in for anything we can’t attribute to measurable deterministic processes. More cynically, it’s a scapegoat when deterministic arguments fail to explain what we see. The implications of changing from a probabilistic definition of stochasticity to one based on the limits to our knowledge are profound. Here, we could make a system less stochastic by measuring more things. I find that last statement incredibly unsettling, and quite contrary to my stochasticity-as-randomness view, but this is where my head starts to hurt. I can usually pretty happily ignore the question of whether anything in ecology is truly random because regardless of the answer, our finite ability to know things will always render ecological systems effectively random. With this thinking, I can’t logically reject the idea that increasing knowledge decreases stochasticity, but the idea still doesn’t fully sit right with me.

Certainly, it matters what we think an ecological pattern is random with respect to. In plant community ecology, for example, the spatial arrangement of different species may be effectively random with respect to one another, but not with respect to fine-scale environmental heterogeneity (or vice versa). Here again, whether a pattern appears to be stochastic or not depends more on whether we’ve measured the right drivers, than on any true random or probabilistic element.

In sum, stochasticity may be randomness, or a probabilistic representation of non-random things we don’t or can’t know. The important distinction between these viewpoints is that in the first, stochasticity is an integral part of the mechanism that generates ecological variation and uncertainty. In the second, stochasticity is something that blurs an inherently predictable pattern, making it only appear uncertain. In the second view, if we could remove stochasticity, we would fully understand the system whereas in the first, stochasticity cannot really be removed nor can the system be fully understood without it.

So, we collectively believe that stochasticity is one or both of these opposing things. There are also several things that we believe stochasticity isn’t. It isn’t variation per se; some ecological variation has deterministic roots, such as population-level variation in demographic rates due to age structure. An individual’s age is deterministic, and so the component of demographic variation due only to age is also deterministic. Stochasticity also isn’t equivalent to neutrality; despite the strong influence of stochasticity on neutral dynamics, stochastic community patterns need not be neutral. Most of us feel that measurement uncertainty and the statistical treatment of error structures in data fall outside the umbrella of stochasticity. (Although I have pondered whether there are hidden biological insights in the way statisticians deal with process error. Folks in my lab meeting didn’t find this particularly compelling, but I will probably continue to ponder, if for no other reason than to build a better conceptual bridge for myself between model-fitting and theory.) Next, stochasticity is not simply the application of stochastic computational tools, like MCMC. Several of us study stochastic processes using deterministic tools, an approach that conserves the idea of a distribution of outcomes without actually sampling from that distribution. Lastly, stochasticity isn’t chaos, although the two might be effectively indistinguishable in a practical sense. (The interaction between stochasticity and nonlinear phenomena like chaos came up a lot in discussion, and is a topic near and dear to my heart and the hearts of many others listed above. To me, this is where the effects of stochasticity get the most interesting: it generates the storage effect, reactivity, and long or recurring transient patterns, for example.)

2) Questions of scale:

I asked my colleagues to explain the origin of stochasticity in ecology, and several pointed to heterogeneity at smaller scales. Heterogeneity of cells within an individual generates stochasticity at the scale of the organism, heterogeneity among individuals generates stochasticity at the scale of the population, and the magnitude of interspecific differences influences the importance of stochasticity at the scale of the community. In other words, deterministic processes that generate heterogeneity at small scales generate variance that is effectively stochastic at larger scales. (Stochastic processes at small scales also surely contribute to stochastic variation at larger scales, but this statement is less interesting.) Conversely, when we measure variance at a particular scale, we tend to attribute it to lower-level processes because, ideally, variation in higher-level processes was controlled in the study design.

As responsible ecologists, we all know the importance of defining the spatial and temporal scale at which a particular idea is relevant. By the same token, I think we should define our “scale of determinism” – the scale outside of which we chalk all variation up to stochasticity, whether or not it’s inherently random. I’m not quite sure how to implement this, but it’s a cute idea. Perhaps the scale of determinism is internally determined: at small scales everything appears random (think of “population dynamics” on a spatial patch so small individuals are continually walking in and out) and at large scales everything appears constant (imagine summing up the population dynamics on a large network of asynchronous local patches). In between is the scale of determinism. Of course, the dynamics at that intermediate scale still won’t be fully deterministic, so perhaps this is a misleading way to think of things. In any case, a “scale of determinism” need not be a spatial scale as in this example; it could be a biological level of organization, or even simply an explicit list of processes (like density dependence) that we consider to be deterministic within the context of a particular study.

3) The term “stochasticity” gives a false sense of precision:

The inherent probabilistic nature means that we can never precisely predict stochastic outcomes. One contributor suggested that this might be why we opt to use the more pretentious-sounding word “stochastic” instead of just saying “random”. At least in common parlance, “random” has connotations of hopelessness. To try to understand something random feels futile; it’s just random. But of course, we can understand many things about stochastic/random processes, despite this inherent unpredictability, by studying the probability distributions.

So, the study of stochasticity lies somewhere between futility and precision. Personally, I’m comfortable in this neighborhood. Still, it’s worth remembering that a solid mathematical description of stochasticity and a biological understanding of stochasticity are two different things. I don’t have any objection to our use of the term “stochasticity”, but I do wonder if we let ourselves get away with muddled terminology in ecology because we rest assured that the mathematicians have sorted it out (“ah yes, ‘stochastic processes’ – people know how to deal with those”). Semantics aside, we do seem to have an awful lot of at least somewhat divergent ideas about stochasticity, even with a hand-picked group of like-minded people.

Closing thoughts: I initiated this blog post, and the conversations that led up to it, hoping to finally get over some of the conceptual issues that have been nagging at me for years. As I wrote, I revised my goal to be to get you, the readers, to feel just as angsty about stochasticity as me. Isn’t it an exciting feeling?! I mean that seriously, no sarcasm: isn’t it exciting? There’s a big, wide, stochastic world out there full of monumentally important questions yet to be answered.

[Bonus material: Chris Stieha took the above text and ran it through a program he wrote that assembled random sentences from clauses in my original text. The result: a stochastic essay on stochasticity. Here are my favorite quotes —

“stochasticity is anything with a hand-picked group of colleagues, collaborators, and lab members”

“stochasticity is an integral part of the term “stochasticity”, but I wouldn’t consider that to be answered.”

“There’s a big, wide, stochastic world out there full of monumentally important questions yet to be deterministic within the context of a distribution of outcomes without actually sampling from this distribution”]

25 thoughts on “A distribution of thoughts around a central tendency: the meaning and impact of stochasticity in ecology”

One pragmatic definition of stochastic is something that is *indistinguishable* from what would occur by chance alone. I know that many proponents of neutral theory use this definition (both implicitly and explicitly) because a good fit between a neutral theory prediction and empirical data does not mean that the empirical patterns were *caused* by neutral mechanisms, just that the consequences of deterministic mechanisms were not obvious from the data.

Another thing that helps me understand stochasticity is the distinction between pattern and process in ecology. Of course, this distinction is a lot harder in practice, but it is still worth considering.

For example, you can get deterministic processes that result in stochastic patterns (deterministic chaos – e.g. May’s work on the population dynamics of a hypothetical species with high growth rate and non-overlapping generations). On the other hand, you can also get stochastic processes that result in deterministic patterns (emergence – e.g. consistent predictions from Hubbell’s neutral theory).

Then you have the conventional understanding found in most basic ecological text books that deterministic processes lead to deterministic patterns. Lastly, you have the stochastic process leading to a stochastic pattern…dispersal of seeds by a turbulent wind, for example.

Just adding a bit to your text above, which may help to give you less of an upset stomach “In sum, stochasticity [is] a probabilistic representation of non-random things we don’t or can’t know [or don’t care to know]”. By “don’t care to know”, I mean that each error (deviation from the model) has a component that is so contingent on a unique history that nothing regular can be learned from the extra knowledge. Think of the simplest possible system: a collection of particles in Brownian motion. The particles are not moving randomly (though they are usefully modeled as such). Each particle has a position and velocity resulting from a history of prior collisions. It is completely deterministic. But no regular knowledge could be gained from knowing this history of collisions.

So even if a model has a low R^2, determinism does NOT necessarily mean that there still remains undiscovered factors creating regular patterns in the data just waiting for a smart ecologist to uncover. It could be a system with a large component of highly contingent factors that nobody cares about.

Fun post to read (thanks!), in part because it echoes a similar process some colleagues and I went through several years ago. Circa 2010, I felt pretty confident that a bunch of smart people meeting once a week to grapple with the idea of stochasticity (specifically in community ecology) would end with clear resolutions to the conundrums you raise. Alas, we had good fun, came up with some interesting perspectives worth sharing (see link below), but fell well short of a grand resolution. A few specifics that speak to some of your points:

(1) I really like the “with respect to” conception of stochasticity (McShea & Brandon 2010, “Biology’s first law”), because it can help a lot with operationalization, legitimately leaving the philosophical question aside (not just pretending it doesn’t exist). My favourite example is this: if the fitness of an individual is random (stochastic) with respect to the allelic states at a particular locus, the result is genetic drift. The fitness of any individual is almost certainly not stochastic with respect to any number of _other_ things, but it may well be stochastic with respect to how many repeats there are in some microsatellite locus. Insert “species identity” for “allelic states” and you have ecological drift; whether or not you think the latter ever happens in nature is unrelated to whether the concept is logically valid.http://mvellend.recherche.usherbrooke.ca/Vellend_etal_Oikos2014.pdf

(2) We ended up with “neutral stochasticity” in our title instead of just “stochasticity”, in part because of point 1. For example, environmental stochasticity (one of the most common flavours) is really a deterministic effect of the environment on fitness that just happens to vary over time. The fact that we can’t predict the environmental fluctuations themselves doesn’t change the fact that the population/community “feels” it in a deterministic way. From the point of view of an ecological community, it was hard to think of flavours of stochasticity that were real but not neutral.

(3) The other expression I like (alluded to in Jeremy’s earlier post) is Sewall Wright’s “practically irreducible probabilities like those in the fall of dice”. Despite deterministic physical laws determining what happens, there is nothing useful (in the context of the game) with respect to which the outcome is not stochastic.

Thanks for pointing to your Oikos paper, Mark (which I actually just re-found on my computer — looks like I downloaded it when it came out but then it slipped off my radar… look forward to reading it for real this time).

Your second point raises a really interesting challenge — is there a type of (environmental?) stochasticity that is “real but not neutral”? I don’t have an immediate answer, but I’m going to keep pondering that… hopefully without having to define “real”… and perhaps after actually reading your paper. But in the meantime, I’m interested to hear others’ thoughts on this.

Perhaps this is just semantics, but I’d argue we can have certainly have environmental stochasticity that is “real but not neutral”. To me, stochasticity is ‘real’ whenever the variable in question is ‘macroscopic’, i.e. it’s dynamics arise from aggregate dynamics at a lower level, which I think covers most of ecology (sure that variation may sometimes be negligible, but no less real).

I’m a bit less clear what we mean by ‘neutral’ in this context, but again I think this depends on what we are talking about (I still have to take a closer look at Mark’s paper; thanks for that link!). ‘Neutral’ may often mean Brownian, but of course we have a nice literature on various types of colored noise. Or perhaps we mean that the average is the same as the deterministic behavior, though we have lots of good examples where this isn’t the case, and stochastic fluctuations have measurable macroscopic / average differences (e.g. inflation/deflation of the mean; though admittedly most of those results just care about the variation of a parameter in time, and not whether it varies in a deterministic or stochastic way). Lastly, ‘neutral’ might mean the noise is ‘non-stationary’, or ‘without drift term’ in the Fokker-Planck equation, which like I think Karen is suggesting, should have plenty of examples in environmental stochasticity.

Hi Karen, this is a problem we’ve struggled with in my lab and it’s nice to know other people are struggling with it. I think the question of ‘is there true stochasticity’ is one of the big questions still to be answered. I ran across some quote from Laplace somewhere (I forget where) that made clear that he thought there was no such thing as true stochasticity. More recently, I remember running across a quote from James Clark that said something like “stochasticity/randomness/error is just all the things we don’t know.” I’m paraphrasing so I may have this badly wrong. I’ve had conversations with physicists who are convinced that Heisenberg’s Principle of Uncertainty (you can never measure perfectly both the speed and location of a particle) means that true stochasticity does exist. Maybe they’re right. But just because there’s uncertainty at the subatomic level – does that necessarily mean there is at all levels. And just because we can’t measure location and speed precisely (using the current state of our technology and imagination) does that mean we’ll never be able to? I don’t know the answer to that. I’m not sure the question has practical implications right now but it certainly has philosophical implications. For example, the famous George Box quote “…all models are wrong but some are useful.” is simply wrong if there is no true stochasticity. (As an aside, I really dislike the way that quote gets used – as if the search for the truth is a fool’s errand – that the only objective should be utility. My guess, is that Box didn’t mean it quite so broadly but it tends to get used that way.). And in theory, if there was no true stochasticity, we could tell when we had perfect knowledge about some specific process – when we could make perfect predictions. If there is true stochasticity then perfect predictions aren’t possible. So, it certainly has philosophical implications. And as we develop better and better models of how the world works it may have practical implications – it may become important to know if there is true stochasticity because that will allow us to know when we have reached the limits of achievable understanding. It’s a coin toss for me – but I probably lean into the Laplace/Clark camp.

Laplace is one of those interesting characters in the history of science; he was very influential in the field of probability and Bayesian interpretations of probability and yet also gave the world the vexing problem of his little demon. Laplace’s all-knowing demon would, if they or some variant existed, show that the entirety of universe was deterministic, and that, as you say, there would be no stochasticity if we had the powers of Laplace’s demon. However I read recently that there wasn’t sufficient computing capacity in the entire universe for Laplace’s demon to work (there hasn’t been sufficient time to allow for the calculations required to know what the demon knows), but even that result is countered by those that push the many-worlds or parallel universe interpretations of quantum mechanics.

From what I recall, randomness is an inherent property at the quantum level – no matter how good our abilities to measure things become, we’ll never be able to know all quantum properties of sub-atomic particles simultaneously. What this means oftens falls into the realm of philosophy and exactly which interpretation of quantum uncertainty one subscribes to, but even the boundary of the scale at which this uncertainty or weirdness can be observed is increasingly being pushed by physicists’ experiments. Whether physicists will ever conclude that quantum effects can be felt at the (relatively) macro level at which we do ecology, I don’t know, but it is quite interesting — and mind-bending — to note how far beyond the scale of electrons and quarks quantum phenomena have been observed in the 100 or so years since the quantum revolution.

To me it doesn’t matter whether stochasticity or randomness is a description of things that are truly random or simply things I’m unwilling to spend enough time on to understand them deterministically. Stochasticity is a statement that I’m going to model something probabilistically rather than deterministically because that is what I want to do (with many factors going into want to do including tractability, relative importance to me, degree of interest to me, my expertise in, etc). In short stochasticity is a modelling approach subject to judgement about what makes a good model.

Hmm…so you disagree with that old post of mine, where I said there are times when this choice isn’t yours to make as a modeler? Giving the example of apparent stochasticity in the phage lysis “decision” that turned out to be a pretty deterministic (and adaptive) function of host cell size?

Perhaps I’m reading something into your phrasing that you didn’t intend? Because it sounds like you think that modeling judgment call often is one of purely personal preference, like a preference for chocolate ice cream over vanilla. So not the sort of thing that others would have grounds to criticize, even if they personally would’ve made a different modeling choice. I agree that it’s often a judgment call, but I think it’s quite possible to make a suboptimal or even flat-out wrong judgment.

I suspect our apparent difference here may be because we have different examples in the backs of our minds? I’m guessing that you have in the back of your mind examples in which we’re trying to build a predictive statistical model for some variable, and have to make a judgment call about bias-variance trade-offs in deciding which or how many predictor variables to include? Whereas in the back of my mind is stuff like that phage work, where the questions of interest are *about* stochasticity. Questions like “Do phage hedge their bets when making lysis decisions?” Or consider a more familiar example: if you want to test for neutral drift as a driver of community dynamics, you are *not* free to just estimate “drift” as “variation in community structure unexplained by whatever environmental and spatial variables I happened to include in my ordination”. You’re not free to make that mistake even if ordination is what you happen to know how to do and even if there’s no other way to study drift with the data you have.

I definitely think there is such a thing as good modelling choices and bad modelling choices. But the same decision (model response to environment as stochastic noise) can be good or bad depending on what my question/goal is and the system.

Well put, Brian — with stochasticity and anything else, there are good modeling choices and bad modeling choices, and the goodness of the choice depends on the question as much as (perhaps more than) the study system.

So, Jeremy, on to your last point: suppose your question dictates that you include stochasticity in your model. Now there are a zillion more modeling choices: which quantities (parameters? state variables? which ones?) become random variables, and what distributions should they have?* If you’re modeling a specific system for which you have data, there’s hope of narrowly answering at least some of these questions through model selection, but what if you’re studying stochasticity in a broader, more theoretical sense? There’s very little consistency out there in how people make these modeling choices, and little general guidance on what makes a particular choice “good” in any given context. I think Brian is right that these choices are often made for convenience, familiarity, preference, etc. Perhaps that’s just fine, but I wish we had a better handle on the consequences of each of these decisions.

Yes, I agree that in the theoretical literature, where people are just studying the effects of “stochasticity”, that a lot of modeling choices get made for reasons of mathematical convenience or because they’re traditional. Particularly regarding which quantities get converted from constants (or quantities that vary in some deterministic fashion) into random variables.

Hi Karen, really interesting piece here and so nice to hear that so many others are also thinking about these questions. (And your own research in this area is also an inspiration to me!) Like you and co-authors, I find that the notion of a stochastic process in ecology is directly a consequence of scale. I’ve always found the comparison to physics of gases (statistical mechanics vs thermodynamics) instructive — we know the air in a room is really composed of atoms bouncing around according to Newton’s deterministic laws, but it would be impossible to compute those for all the 10^23 or so atoms. Writing it down as a stochastic process works out really well, not only in predicting the mean dynamics (wherein we recover the thermodynamic laws like PV=nRT discovered long before we had a notion of atoms), but can also predict probability of deviations of a given size (best observed under extreme vacuum conditions where there are fewer molecules). These calculations usually assume all the atoms are identical, when of course we know air is really made up of different species of molecules (a familiar assumption for us in theoretical ecology as well). In fact, we can go back and add this in, accounting for the different dipole moments of different molecules, and predict deviations from the ideal gas law that can be confirmed by observation (again only in cold or rarified gases, if I’m remembering college homework assignments correctly). Thus, concepts like “Temperature” or “Pressure” are inherently stochastic ideas, not because there is anything fundamentally probabilistic and non-deterministic (no quantum-weirdness enters here) about them, but because the deterministic laws giving rise to them operate on a different scale. Temperature or pressure is by definition an average over that collection.

I think the same is true of almost everything we consider in ecology — we discuss concepts like the dynamics of genes or species abundances which are really aggregations, averages, arising out of deterministic processes at a lower scale (or approximately deterministic — it’s turtles all the way down). We tack a bit of stochasticity on the end of our equations to reflect that our variable of interest (density of a species, say) is really a ‘macroscopic’ quantity like temperature that arises from lower-level processes. If we’re more pedantic about it we can calculate this explicitly, and compute deviations (e.g. our SDEs like dX_t = f(X_t) dt + g(X_t) dBt are only approximately accurate), as we see in stuff like the master equation stuff that often pops up in our literature.

“We tack a bit of stochasticity on the end of our equations to reflect that our variable of interest (density of a species, say) is really a ‘macroscopic’ quantity like temperature that arises from lower-level processes” — Indeed, very well said. I have a love/hate relationship with this approach (which I use all the time, by the way, so apparently “love” is winning out thus far), because I find the arguments in favor of instead adding stochasticity to the lower-level process (or the parameters that represent it — e.g. population growth rate, representing individual births/deaths in aggregate) quite compelling. The downside is, taking this approach amplifies the number of modeling choices that need to be made (see Brian’s comment and replies above), where instead of summarizing all lower-level variability with one ‘macroscopic’ probabilistic distribution, we need a distribution for each process. It isn’t clear that this added level of complexity is worth much, except in very well-studied systems where we have good information on what all these distributions should be.

Anyway, with that debate as a backdrop… I saw Jeff Gore give a talk a year or two ago, in which he presented this work:http://www.gorelab.org/Science-2012-Dai-1175-7.pdf
on yeast populations that appear to undergo a fold bifurcation as their mortality rate is experimentally altered. The purpose of the study was to test leading indicators of the ecosystem collapse that occurs once the populations cross this bifurcation. These indicators are based on how a system responds to perturbations at different distances from the bifurcations (go read Carl’s papers if you want to learn more), and so the experiment involved perturbing these yeast populations by adding a pulse of something that kills them (sodium, I think). As Jeff was explaining this, he showed a bifurcation diagram, with mortality rate on the x-axis and yeast population density on the y-axis, and he drew a big down arrow on the graph to represent the perturbation, indicating that the salt shock reduced population size.

At the end of the talk, I asked Jeff why he didn’t draw the perturbation as a sideways arrow, indicating that the salt shock increased mortality rate. He said he could’ve drawn it either way and it didn’t really matter… which was true, as I recall, in the context of his study, but to me, this is exactly the debate I described at the start of this reply! Should we think of a perturbation (a stochastic one, or an experimental shock) as a bump to the lower-level process (e.g. births/deaths) or a bump directly to the macroscopic variable of interest (e.g. population density)? I had always thought of this as a theoretician’s problem — one that only arises when one (as I do) tries to represent general phenomena in general models. But here was someone with a REAL SYSTEM, highly controlled, and where the nature of the perturbation is known exactly — and even he couldn’t say whether the perturbation had been to the death rate or to the population size. I found this both gratifying and depressing.

Karen, really interesting case in point! I’m very surprised/confused about Jeff’s answer as well. They have an immensely elegant experimental system there, in which they have, if I recall correctly, three different ‘drivers’ or knobs they can tune, any of which can cause the bifurcation to occur: serial dilution percentage they use in their transfers (the one in the Science paper, which they sometimes call death rate), nutrient concentration, or salt shocks (see their PNAS paper on this: http://www.gorelab.org/PNAS-2015-Dai-1418415112.pdf, in which they cleverly figure out why they couldn’t really see the signal in the latter two knobs).

If he had ‘death rate’ on the x axis and ‘population density’ on the y axis, I’m curious if he meant ‘death rate’ as a parameter determined by the salt concentration, or rather as determined by the serial dilution (diluting a population having, of course, the same impact on population density as death does, but without any dying involved). If that was the case, and then salt was introduced as a sudden, large shock, then absolutely I would say the arrow should be a down arrow, not horizontal — the dilution regime didn’t change, but an external factor (salt) had lowered the population size.

If instead he was considering the bifurcation diagram where dilution is constant, but salt is the ‘bifurcation parameter’ on the x axis that is (usually slowly, continuously) increasing, then the arrow would have to be horizontal, since the x-axis was literally the salt concentration, and a down arrow would be wrong.

I don’t see any way in which the answer could be “either is okay”. Notably, they have both different units and different magnitudes to be effective: a shock to population density, y axis, only results in a transition if it’s big enough to cross the separatrix, while a shock to salt concentration (if salt concentration is the horizontal axis) results in a transition only when it passes the tipping point.

Interestingly, I don’t think either of these is very carefully defined in terms of a low-level process. As much as I am huge fan of the work coming out of that group, (which appears most rigorous in both experimental and theoretical aspects), the population model they use in these papers is one I find deeply unsatisfying, having neither very precise ties to the experimental variables (salt concentration, nutrient concentration etc) nor the kind of emergent multi-stable states we usually enjoy deriving from non-linear models. Here was my go at playing with their model: http://www.carlboettiger.info/2015/02/12/dai-model.html . As you see, the model ‘hard-wires’ a step-like change in growth regime at a fixed concentration. Normally, we would expect this to emerge by itself whenever we have the positive/cooperative/allee interactions at low density which these yeast clearly do have (cooperatively breaking down sugar). I really hoped to at least see a positive x^2 term if not a lower-level stochastic model… Like you say, gratifying and depressing that we can all (even MIT physicists who are also excellent empiricists) struggle over modeling choices!

Ha! Yes, Carl, I had a very similar thought (i.e. that the correct direction of the arrow depends on exactly what mortality parameter is on the x-axis, and that it’s hard to imagine how either kind of arrow could be equally right) while I was writing all that yesterday, but I tried to get away with brushing it under the rug. The truth is, I don’t remember the details well enough to justify this now, but I do remember thinking in the moment that Jeff’s “it doesn’t matter” answer made a frustrating amount of sense. So either it’s true that this is an example where there are multiple, supportable mathematical representations for the same perturbation; or Jeff’s off-the-cuff “it doesn’t matter” answer was incorrect and I didn’t see it at the time… wish I could say more about which I think is the more likely. Even if Jeff’s answer was incorrect, it’s interesting to me that the proper representation wasn’t immediately obvious in a case like this.

I never took the time to look at their model in close detail, and your exploration of it sounds really interesting so I’ll go take a look at it – thanks!

Thanks for the links! The Pascual & Levin paper was absolutely the inspiration behind some of my comments on scale, although until I saw your comment, I hadn’t been able to put my finger *where* I had read that stuff many years ago. Someone at my lab meeting also mentioned seeing a similar concept presented in the physics literature.

I’m not familiar with the Keller paper and look forward to giving it a read.

From the ‘stochasticity is just unexplained information’ camp, are stochastic effects then measured from the residuals of a model? Presumably with this philosophy, what one person would call stochasticity is only a problem of missing covariates.

Really interesting subject. Do wish I had a stronger mathematical base to delve deeply enough to form thoughts with less hand waving. 😉

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